Question

Write each of the following sets by listing their elements between braces. $$ \\{3 x+2: x \in \mathbb{Z}\\} $$

Short answer

The set \(\{3x+2 : x \in \mathbb{Z}\}\) is the set of numbers obtained by multiplying any integer by 3 and then adding 2. Therefore, there are infinitely many elements in this set.

Step-by-step solution

Understand the given set notation

The set is \(\{3x+2: x \in \mathbb{Z}\}\). This notation means \(3x+2\) for all \(x\) in the set of integers, denoted by \(\mathbb{Z}\).

Apply the operation to integer values

Substitute some integer values into \(x\) in the equation \(3x+2\). For instance, when \(x=-2, -1, 0, 1, 2\), the results are \(3(-2)+2=-4\), \(3(-1)+2=-1\), \(3*0+2=2\), \(3*1+2=5\), and \(3*2+2=8\) respectively.

List all elements in the set

Knowing that the values obtained are the outcome for only these particular values of \(x\), we can say that the elements of the set \(\{3x+2: x \in \mathbb{Z}\}\) are all numbers of the form \(3x+2\) where \(x\) is an integer. Since all integers can be used, the pattern continues both positively and negatively without end, and there are infinitely many elements in the set.

Key concepts

Understanding Integer Sets

The concept of an 'integer set' is foundational in mathematics and represents a collection of numbers consisting of whole, non-fractional values that can be positive, negative, or zero. An integer set is symbolized by the notation \( \mathbb{Z} \). This set encompasses numbers such as -3, -2, -1, 0, 1, 2, 3, and so on, extending infinitely in both directions on the number line.

When working with integer sets in exercises, you are often asked to perform various operations on these sets or to understand their properties. Integer sets are crucial because they allow for a simplified form of arithmetic and algebra, and they are part of many mathematical structures and theorems. Being comfortable with integer sets lays the groundwork for further studies in number theory, algebra, and beyond.

Performing Mathematical Operations

Mathematical operations such as addition, subtraction, multiplication, and division are fundamental actions performed on numbers. In the context of integer sets, these operations can be applied to each member of the set. The exercise \( \{3x+2: x \in \mathbb{Z}\} \) is an example of a mathematical operation—specifically, an algebraic expression—applied to an integer set.

In this case, the operation is 'multiply by 3 and add 2'. This combination of multiplication and addition is a reflection of how we can apply operations sequentially to generate new sets of integers. When approaching problems involving operations on sets, it's crucial to perform these operations methodically, ensuring the result is in the proper form and remains within the set's structure.

Element Listing in Sets

Listing elements of a set is a way to explicitly demonstrate the members of that set. It is common in mathematics to describe a set by the properties that its elements must satisfy, and then, if possible, to list out its elements if the set is finite or if only a portion of the set is being considered.

In exercises where you list the elements of a set, as in our example, \( \{3x+2: x \in \mathbb{Z}\} \) you usually provide several instances to showcase the pattern within the set. For this particular set, every integer \(x\) is transformed by the algebraic operation \(3x+2\). The resulting values are the elements of the new set. It's helpful to write down a few examples to see the trend—negative integers will result in negative or zero values, and positive integers will yield positive values. Element listing in sets is not only a representation tool but also a way to understand the underlying patterns and structures of the sets involved.